Part VI · Information & Entropy — 15

The Holevo Bound

A qubit’s state is a point on a globe — uncountably many possibilities. It is tempting to think you could hide a whole encyclopedia in one. You can’t. Holevo’s bound caps what you can ever read back out at a meagre one bit.

↩ before you start · keep these handy

·From Ch. 3: measuring a qubit returns just one click and collapses everything else.

·From Ch. 13: a single qubit’s entropy S(ρ) can never exceed 1 bit.

·From Superdense: “two bits per qubit” only works by also spending a pre-shared entangled qubit.

🔑 symbol decoder · every new mark, in plain words

χ"chi" — the Holevo cap: the most classical info any measurement can pull out. ρ̄the average state — Alice’s encodings blended by how often she sends each. pᵢ, ρᵢthe probability of, and the state for, message i. overlapcos(φ/2) — how similar two states are; 0 = opposite poles, 1 = identical. accessible infowhat Bob actually learns from his best possible measurement.

feel

Infinite to write, one bit to read

You can prepare a qubit in any of infinitely many states, choosing an angle to absurd precision. But to get information back you must measure — and measurement (chapter 03) gives only one of two clicks, then destroys the rest. Worse, if your chosen states overlap, even those clicks are ambiguous. The readout is a narrow doorway, and Holevo measured exactly how narrow.

🎵 everyday picture

Two tuning forks. If their pitches are far apart, your ear tells them apart instantly — one clean bit of information. Tune them closer and closer and the difference becomes a faint wobble you can’t reliably name; eventually you’re just guessing. The forks still differ physically, but your one listen can’t resolve it. Holevo’s bound is the math of exactly how much your single “listen” (a measurement) can ever distinguish.

recapYou can write a qubit with infinite precision, but reading is a single narrow doorway — Holevo measures its width.

play

Two states, one noisy answer

Alice encodes one bit by sending one of two equally-likely pure states, ψ₀ or ψ₁, separated by an angle on the Bloch ball. Widen the angle and they grow distinguishable; narrow it and Bob can barely tell them apart. The bars compare what Bob can actually read, the Holevo cap χ, and the hard 1-bit ceiling.

▸ accessible information ≤ χoverlap = cos(φ/2)

Bob reads (Helstrom){{ accBits }}
Holevo cap χ{{ chiBits }}
1-qubit ceiling1.000

              angle φ
              
            
overlap |⟨ψ₀|ψ₁⟩|{{ overlap }}
readout error{{ perr }}
{{ desc }}

recapThe more the two states overlap, the smaller χ — only opposite poles let a full bit through.

math

The Holevo quantity χ

Alice sends state ρ_i with probability p_i. The information Bob can extract by any measurement is capped by χ — the entropy of the average state minus the average of the entropies:

accessible info ≤ χ = S(ρ̄) − Σi pi S(ρi)
pure encodings:  S(ρi) = 0, so  χ = S(ρ̄) ≤ 1 bit for one qubit
overlapping states:  ρ̄ stays near the surface → small S → tiny χ

Because a single qubit’s entropy S(ρ̄) can never exceed one bit (chapter 13), χ can’t either. Only when the encoding states are orthogonal — pushed to opposite poles — does the average state reach the center, S = 1, and a full bit become readable.

✎ worked example · two pure states 90° apart

each encoding is pure, so S(ρᵢ) = 0 — the second term vanishes, χ = S(ρ̄).
average state length r = cos(φ/2) = cos45° = 0.71 (it sits inside the ball, not orthogonal).
eigenvalues (1±0.71)/2 = 0.85 and 0.15 → S(ρ̄) = −0.85log₂0.85 − 0.15log₂0.15.
χ ≈ 0.60 bits — below 1. To read a full bit Alice must push the states to 180° (opposite poles). ✓

recapχ = S(ρ̄) − Σ pᵢ S(ρᵢ); for one qubit it can never top 1 bit.

⚠ common misconception

“A qubit holds a continuous amplitude, so it stores infinite classical data.” It can be prepared with infinite precision, but measurement is the only way out, and Holevo caps the readout at one bit per qubit. The continuum is real but unreadable — superdense coding’s “two bits per qubit” doesn’t break this, because it spends a second, pre-shared qubit of entanglement.

With this, the information story is whole: a qubit stores S qubits’ worth (chapter 14) and reveals at most one classical bit. Entropy has set both the floor and the ceiling — the natural close of Part VI.

✓ you can now

✓explain why a qubit can be written with infinite precision but read out at only one bit

✓compute the Holevo cap χ = S(ρ̄) − Σ pᵢ S(ρᵢ) for pure overlapping encodings

✓say why superdense coding doesn’t break the bound — it spends shared entanglement

← 14 Data Compression end of Part VI · roadmap →